Open Sets Avoiding Integral Distances

Abstract

We study open point sets in Euclidean spaces \(\mathbb{R }^d\) without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density \(0\). We are interested in how large such sets can be in \(d\)-dimensional volume. We determine the exact values for the maximum volumes of the sets in terms of the number of their connected components and dimension. Here techniques from diophantine approximation, algebra and the theory of convex bodies come into play. Our problem can be viewed as a counterpart to known problems on sets with pairwise rational or integral distances. It possibly opens a new research direction with strong links to topology and measure theory.